Dynamical charged black hole spontaneous scalarization in anti–de Sitter spacetimes

We study the fully nonlinear dynamics of spherically symmetric black hole spontaneous scalarization in Einstein Maxwell scalar theory with coupling function $f(\ensuremath{\phi})={e}^{\ensuremath{-}b{\ensuremath{\phi}}^{2}}$, which can transform usual Reissner-Nordstr\"om anti--de Sitter (RN-AdS) holes into hairy holes. Fixing Arnowitt-Deser-Misner mass system, initial perturbation will destroy original RN-AdS and turn it a provided that constant $\ensuremath{-}b$ charge are sufficiently large, while cosmological is small enough. In process, we observe irreducible initially increases exponentially, then approaches to finally saturates at finite value. Choosing stronger larger charge, find exponentially grows earlier takes longer time for be developed stabilized. further examine phase structure properties process confirm observations dynamical study.